Math Is Fun Forum

GOKILL

Member

Registered: 2010-03-19

Posts: 26

Abstract Algebra - Ring [extended level] =_="

(1)
Given Integral domain R and

a)

not unit, is called prime element
if a|bc implies a|b or a|c.
Show that a is prime element iff <a> is prime ideal
b)

not unit, is said un-reduced
if a = bc implies b is unit or c is unit.
Show that if p is prime and

for some i.

(2)
If

. What criteria must be held by

? (is it prime or un-reduced?)

Anyone can help me plz....

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JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Abstract Algebra - Ring [extended level] =_="

(1) Use induction.

(2) Hint:

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Abstract Algebra - Ring [extended level] =_="

For (1a), remember the phrase "to contain is to divide".  If (a) contains (b), then a divides b.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."